Worksheet 1 (1



Worksheet 1 (1.1)

Chapter 1 Basic Concepts and Properties

1.1 Sets, Real Numbers, and Numerical Expressions

Summary 1:

An algebraic expression is a generalization of arithmetic ideas using variables to represent any number.

Warm-up 1. a) The sum of a and b is written .

b) The quotient of c and d can be written .

c) The product of x and y can be written .

Problems

1. Write "the difference of c and x" in algebraic form.

2. Write "the product of 5 and z" in algebraic form.

Summary 2:

A set is a collection of objects.

These objects are called elements or members.

Braces, { }, are used to enclose sets.

Sets are named with capital letters.

The null set, ∅, is a set containing no elements.

If a pattern is established when listing a set, 3 dots can be used to show the pattern continues. {1,2,3, . . .}

∈ is read "is an element of"

∉ is read "is not an element of"

Equal sets have the exact same elements.

Warm-up 2. a) If A = {the last three letters of the alphabet}, represent set A as a list. A = { }.

b) If B = {March, May}, represent set B in words.

B = { }.

c) If C = {2, 4, 6, 8, . . .}, represent set C in set builder notation.

C = {x| }.

Worksheet 1 (1.1)

d) If D = {x|x is a day of the week beginning with the letter P}, then set D = .

e) If A = {a,b,c,d, . . .}, is h ∈ A? .

Problems

3. Write A = {the first 3 letters of the alphabet} in listing form.

4. Write B = {3, 6, 9, 12, . . . } in set-builder notation.

5. Is 15 ∈ B?

Summary 3:

Sets of Numbers

natural numbers, counting numbers or positive integers: {1,2,3,4, . . .}

whole numbers, or nonnegative integers: {0,1,2,3, . . .}

negative integers: {. . . -3,-2,-1}

nonpositive integers: { . . . -3,-2,-1,0}

integers: { . . . -3,-2,-1,0,1,2,3, . . . }

rational numbers: {a/b| a and b ∈ Integers, b ≠0} EX. 2/3, -4, 0, .3

(A rational number can also be defined as any number which can be written as a terminating or repeating decimal.)

irrational numbers: {numbers that cannot be written in a/b form}. EX. π

(Irrational numbers will not terminate or repeat as decimals.)

real numbers: {a number which is either rational or irrational}

A is a subset of B (A ⊆ B ) if every element of A is also an element of B.

A _ B is used to indicate that A is not a subset of B.

Warm-up 3. a) The number 5/3 is an element of which sets listed above?

b) The number -8 is an element of which sets listed above?

c) Is the set of integers a subset of the set of natural numbers?

Problems

6. Is 2/3 ∈ the set of integers?

7. Is -5 ∈ the set of rational numbers?

8. Identify every set to which the number -1/2 belongs.

Worksheet 1 (1.1)

Summary 4:

Properties of Equality, for any real numbers a, b, c

1. Reflexive Property: a = a

2. Symmetric Property: If a = b, then b = a

3. Transitive Property: If a = b, and b = c, then a = c

4. Substitution Property: If a = b, then each can replace the other.

Warm-up 4. Complete the right side to illustrate the property listed:

a) x + y = (Reflexive property)

b) If x = 15 and x - y = 9 then - y = 9 (Substitution Property)

c) If x = 5 and 5 = a then (Transitive Property)

Problems - Complete the right side to illustrate the property listed:

9. If x = y, then (Symmetric Property)

10. If x = 8, and 8 = y, then (Transitive Property)

Summary 5:

Order of Operations

Operations on numerical expressions will be performed in the following order:

1. Expressions contained in grouping symbols. ( ), { }, [ ], ----, √

(Begin with the innermost set of symbols.)

2. Evaluate any exponents.

3. Multiply or divide in the order of appearance from left to right.

4. Add or subtract in the order of appearance from left to right.

Warm-up 5. Simplify:

a) 5 + 6 - 3⋅4 ÷ 2 = 5 + 6 - ÷ 2

= 5 + 6 -

= -

=

b) 3 + 2[9 - 2⋅3] = 3 + 2[9 - ]

= 3 + 2[ ]

= 3 +

=

Worksheet 1 (1.1)

c) [pic] =[pic]

=[pic]

=[pic]

=

Problems - Simplify:

11. 16 - 12 ÷ 4 ⋅ 2 + 3

12. 12 - 2[15 - 3⋅4]

13. 18 - [5(3 - 1)] + 5

14. [pic]

Worksheet 2 (1.2)

1.2 Operations with Real Numbers

Summary 1:

The symbol "-" can be read negative, opposite or additive inverse.

The opposite of a positive number is a negative number: -(5) = -5

The opposite of a negative number is a positive number: -(-8) = 8

The opposite of zero is zero: -(0) = 0

Double negative property: -(-a) = a

> represents greater than; ≥ represents greater than or equal to.

< represents less than; ≤ represents less than or equal to.

Warm-up 1. a) -(-12) =

b) "the opposite of x" would be written algebraically as .

c) If -2 is to the left of 5 on a number line, then -2 5. (< or >)

Problems

1. -(-25)=

2. the opposite of 5 is

3. Place a > or < between the two numbers -5 -8.

Summary 2:

Absolute Value

Geometric definition: The absolute value of any number, denoted by vertical bars, | |, is the distance the number is from zero on the number line.

Algebraic definition: For all real numbers a, |a| = a if a ≥ 0 and

|a| = -a if a < 0. This should be read "the absolute value of a is the opposite of a if a is less than zero".

Warm-up 2. a) |16|= c) |0|=

b) |-20|= d) -|-9|=

Problems - Simplify:

4. -|31| 5. |-10| 6. |5|

Worksheet 2 (1.2)

Summary 3:

Addition of Real Numbers

1. To add two positive real numbers, add the two numbers and keep the positive sign.

2. To add two negative real numbers, add their absolute values and keep the

negative sign.

3. To add a positive real number and a real number, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Warm-up 3. a) 8 + 10 =

b) (-5) + (-13) = - (|-5| + |-13|)

= - ( + )

=

c) 15 + (-3) = (|15| - |-3|)

= ( - )

=

d) 17 + (-30) = - (|-30| - |17|)

= - ( - )

=

e) -14 + 0 =

f) -20 + 20 =

[pic]

Problems - Simplify:

7. 23 + (-15) 8. 0 + (-13) 9.

10. (-9) + (-15) 11. 29 + (-29)

Summary 4:

Subtraction of Real Numbers

If a and b are real numbers, then a - b = a + (-b)

(Change subtraction to add the opposite and follow the rules for addition.)

Worksheet 2 (1.2)

Warm-up 4. a) 10 - 15 = 10 + (-15) =

b) (-12) - (-3) = (-12) + (+3) =

c) 21 - 18 = 21 + (-18) =

d) -16 - 3 = -16 + (-3) =

e) 19 - 15 + 8 - (-5) - 3 = 19 + (-15) + 8 + (+5) + (-3)

= + 8 + 5 + (-3)

= + 5 + (-3)

= + (-3)

=

Problems - Simplify:

12. 29 - (-14) 13. (-7.4) - 9.2

14. -11- (-4) 15. 32 - 40

16. -8 - (-3) + 5 - 12 + 9 17. (4 - 6) - (9 - 5)

Summary 5:

Multiplication of Real Numbers

1. The product of two real numbers with the same sign is the product of their absolute value.

2. The product of two real numbers with different signs is the opposite of the product of their absolute values.

3. The product of zero and any integer is 0.

Warm-up 5. a) (-10)(-3) = |-10|⋅|-3|

= ⋅ =

b) (5)(-7) = - (|5|⋅|-7|)

= - ( ⋅ ) =

c) (-11)(0) =

Worksheet 2 (1.2)

[pic]

Problems - Simplify:

18. (-15)(2) 19. (-30)(-6) 20.

21. (0)(-3) 22. (12)(-4)

Summary 6:

Division of Real Numbers

1. The quotient of two real numbers with the same sign is the quotient of their absolute values.

2. The quotient of two real numbers with different signs is the opposite of the quotient of their absolute values.

3. The quotient of zero and any nonzero real number is zero.

Warm-up 6. a) [pic]=[pic]=[pic]=

b) [pic]

c) [pic]

d) [pic]

Problems

23. [pic] 24. [pic]

25. [pic] 26. [pic]

Warm-up 7. Use the four basic operations to simplify:

a) -2 - 5(3) - (2)(-6) = -2 - -

= -

=

Worksheet 2 (1.2)

b) 5 - 3[2(-3 - 8)] = 5 - 3[2( )]

= 5 - 3[ ]

= 5

=

Problems - Use the four basic operations to simplify:

27. 3 - 2(-4) + 3(5)

28. -2 + 4[12 - 5⋅4]

Worksheet 3 (1.3)

1.3 Properties of Real Numbers and the Use of Exponents

Summary 1:

Properties of Real Numbers, for all a, b, c ∈ Reals

Closure Property of Addition: a + b is a unique real number

Closure Property of Multiplication: ab is a unique real number

Commutative Property of Addition: a + b = b + a

Commutative Property of Multiplication: a⋅b = b⋅a

Associative Property of Addition: (a + b) + c = a + (b + c)

Associative Property of Multiplication: (ab)c = a(bc)

Identity Property of Addition: a + 0 = 0 + a = a

Identity Property of Multiplication: a(1) = 1(a) = a

Additive Inverse Property: a + (-a) = (-a) + a = 0

(-a is called the opposite of a)

Multiplicative Inverse Property: [pic]

(1/a is called the multiplicative inverse or the reciprocal of a.)

Distributive Property: a(b + c) = ab + ac

Multiplication Property of Zero: x(0) = 0(x) = 0

Multiplication Property of Negative One: a(-1) = (-1)a = -a

Warm-up 1. Identify the property that justifies each statement:

a) 8(x + y) = 8x + 8y

b) 0(40) = 0

c) [pic]

d) 5ab = 5ba

e) x + (y + 2) = (x + y) + 2

f) x + 0 = x

g) -1(5) = -5

Worksheet 3 (1.3)

Problems - Identify the property that justifies each statement:

1. 5(6⋅4) = (5⋅6)4

2. -1(a) = -a

3. ½ + (-½) = 0

4. 3(a - b) = 3a - 3b

5. x + 5 = 5 + x

Warm-up 2. Simplify, using the properties when convenient:

a) 15(32) + 15(18) = 15( + )

= 15( )

=

b) (½)(32)(20) = ( )( )(32)

= ( )(32)

=

c) 15 + (-3) + (-8) + 5 + (-22) + 6 = 15 + 5 + 6 + ( ) + ( ) +( )

= 26 + ( )

=

Problems - Simplify, using the properties when convenient:

6. (15 + 39) + (-39)

7. 5(19) - 5(11)

8. (-7) + 10 + 8 + (-13) + (-15) + 7

Summary 2:

In the expression bn, b is called the base and n is called the exponent.

This expression can be read "b to the nth power".

If n is a positive integer and b is any real number, then bn means to multiply b times itself n number of times. bn = b⋅b⋅b⋅ ⋅ ⋅b

n factors of b

If n = 2, then b2 can be read "b squared".

If n = 3, then b3 can be read "b cubed".

Worksheet 3 (1.3)

Warm-up 3. Simplify:

a) 43 = ⋅ ⋅

=

b) (_)2 = ( )( )

=

c) (-2)4 = (-2)(-2)(-2)(-2)

=

d) 23 - 52 = -

=

e) [-2(4) + 3(-2)2]3 = [ -2(4) + 3( )]3

= [ + ]3

= [ ]3

=

Problems - Simplify:

9. (¾)3

10. (-4)2

11. -42 - (-3)2

12. -2(.5)2 + 4(.2)3

13. (-1)3- 2(3)2(-1) - 5

14. [5(-2)2 - 3(4)2 + 18]2

Worksheet 4 (1.4)

1.4 Algebraic Expressions

Summary 1:

Terms are algebraic expressions such as 3xy2.

Literal factors are the variables involved in a term. EX. x, y

The numerical coefficient is the numerical factor of the term. EX. 3

Similar terms or like terms are terms that have the same literal factors.

EX. 5a2b and -8a2b

To combine similar terms use the distributive property to combine the coefficients of the like terms.

Warm-up 1. Simplify by combining similar terms:

a) 8x + 5x = ( + )x

=

b) 8a - 3 - 4a + 2 = 8a + ( ) + (-3) +

=

Problems - Simplify by combining similar terms:

1. 18a - 6b + 2a + 5b

2. 2x2 - 3x + 6x - x2

Warm-up 2. Simplify by removing parentheses and combining similar terms:

a) 2(x - 5) + 3(x + 6) = 2x - 10 + +

= 2x + + (-10) +

=

b) 3(a - 4) - (2a + 5) = 3a - - 2a -

=

Problems - Simplify by removing parentheses and combining similar terms:

3. -2(y - 3) + 6(2y + 5)

4. 5(x + 2y) - 6(3x - y)

Worksheet 4 (1.4)

Summary 2:

Evaluating algebraic expressions is the process of finding the value of an algebraic expression when each variable in the expression is replaced by a real number.

Warm-up 3. a) Evaluate 5y - 6z for y = -2 and z = 3.

5y - 6z = 5( ) - 6( )

= -

=

b) Evaluate 3xy2 - 5x2y + 6x3 for x = -1 and y = 3.

3xy2 - 5x2y + 6x3 = 3( )( )2 - 5( )2( ) + 6( )3

= 3( )( ) - 5( )( ) + 6( )

= - +

=

Problems

5. Evaluate 8a - 3b for a = -2 and b = 3.

6. Evaluate x2y - 3xy2 + 2xy for x = -1 and y = 3.

7. Evaluate (2x - 3y)(x - 5y) for x = 2 and y = -4.

Warm-up 4. Simplify the expression, then evaluate for x = -2 and y = 5:

a) 5x - 9y + 3x + 2y = x - y

= (-2) - (5)

= -

=

Worksheet 4 (1.4)

b) 3(5x - 1) - 4(x + 3) = 15x - 3 - -

= x -

= (-2) -

= -

=

Problems - Simplify the expression, then evaluate for the given values:

8. 5a - 6b + a + 2b; a = 7, b = -4

9. -2(x - 5) + 3(2x + 4); x = - ½

Summary 3:

Translating from English to Algebra

Phrases that indicate addition: sum, more than, plus, increased by, added to

Phrases that indicate subtraction: minus, less than, decreased by, difference, subtracted from

Phrases that indicate multiplication: times, product, of, twice, multiply

Phrases that indicate division: quotient, divided by, ratio

Warm-up 5. Translate into an algebraic expression using x to represent the unknown number:

a) six more than a number: +

b) the product of ten and a number: ( )( )=

c) the quotient of a number and fifteen: ( ) ÷ ( ) or [pic]

d) the difference between eight and a number: -

e) nine subtracted from the product of five and a number: ( )( ) -

f) five times the sum of two and a number: 5( + )

Worksheet 4 (1.4)

Problems - Translate into an algebraic expression using y to represent the unknown number:

10. The ratio of a number and six.

11. A number increased by twelve.

12. Twice a number decreased by four.

13. Five more than the quotient of a number and two.

Warm-up 6. Answer the question with an algebraic expression:

a) Ryan is n years old. How old was he 5 years ago? -

b) If n represents an integer, represent the next larger integer. +

c) The sum of two numbers is 45 and one of the numbers is x.

What is the other number? -

d) The cost of 10 pounds of cat food is x dollars.

What is the price per pound? ÷ or [pic]

Problems - Answer the question with an algebraic expression:

14. Mark is x years old and his mother is one year more than three times as old as Mark. What is the age of Mark's mother?

15. Heather has d dimes and q quarters in her wallet. How much money in cents does she have?

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